4 ms·

The rub is the thinking a length of 4 maps to four points when in reality, the points are 4,3,2,1,0, totaling 5. I feel like this could all be helped if in casu

The rub is the thinking a length of 4 maps to four points when in reality, the points are 4,3,2,1,0, totaling 5. I feel like this could all be helped if in casual counting we started at zero, then our entire concept of where the measurements start would be more in line with math. I think often about these fundamental conflicts in how we casually think about numbers and how they are actually modeled in math

CRConrad 2y ago
> I feel like this could all be helped if in casual counting we started at zero So when you've counted to one you've actually counted to two? I feel like some people have been damaged by C-like languages.
aaplok 3y ago
To count the length of the segment, we count the number of unit segments that compose it. Start with 1 (the first segment) and count up to 4 (the fourth segment), yielding a length of 4. We're consistent with the units: we are adding up lengths to get a length (metres, yards, or whatever). Counting (ie, adding up) points gives a number of points, which isn't a unit for lengths. Starting the enumeration from zero is a hack to recover the previous process of adding up unit lengths. This hack only works in this specific context of conversion. If you want to count points (say, the number of corners in the triangle) you'd need to start from 1.
sparky_z 3y ago
I'm sorry, but your suggestion doesn't make any sense to me. Saying "I have zero coconuts" would mean that you have a coconut? Would you have to say you had "negative 1" coconuts if you didn't have any? At a sandwich shop you would have to order a "no feet long" sub? And this is your plan to prevent confusion? All that would really do swap the meaning of a bunch of words around so that "zero" means "one", "one" means "two", etc. Then we'd have to call it a "Two three four" right triangle but remember to make it with "three" (four), "four" (five), and "five" (six) stones and we're right back where we started. The problem here is confusion about what quantity is actually being counted: the "fence posts" or the "fence lengths"? That's always going to depend on context - the speaker and listener have to be on the same page. There's no way to fix that by changing the number we count from.
- baerrie 3y ago
  Where do I say anything about zero becoming 1? What I am talking about is that counting arbitrary periods in a given quantity always starts with zero whether you count it or not. If you ground all of those coconuts up and decided to count how many cups of coconut you have in that mass, well you would be starting with an empty cup measure, which is zero. It’s not just fence posts
  - lupire 3y ago
    Lok up "ordinal" and "offset". They are two different concepts that cannot be merged. Always using offsets will just cause confusing elsewhere.
    - YeGoblynQueenne 2y ago
      It's "ordinal" and "cardinal" really. If you wanna be all mathsy and stuff.
      - YeGoblynQueenne 2y ago
        Sorry about the above comment. I'm being way too snarky in this thread. I hope the following makes up for the snark in my other comments. What I meant above is that, in set theory, there are ordinal numbers and cardinal numbers, both of which are natural numbers. The easiest way to understand them is in terms of arrays: the number of elements in an array is a cardinal whereas an index into the array is an ordinal. Ordinals start at 0, because that's the first natural number, and that's prooobably? why array indexing traditionally begins at 0 (I'm totally guessing). Cardinals also start at 0. So, 0 is the "cardinality" of the empty set, {}, which has no elements and so no ordinal. In the set {+}, 0 is the ordinal that corresponds to the position of the single element of the set, in the set {+,+} 1 is the ordinal for the last element in the set etc. How this is relevant to the article above is that the artist very clearly intended the number of nuts in the "squares" at the sides of the triangle to be understood as cardinals: the squares represent sets and the cardinality of each set corresponds to the surface of that square. Sets are not ordered so the artist is free to place their elements in any arrangement, including a square, and placing a set in the form of a square on the side of an edge of the triangle clearly signals that its cardinality corresponds to the square of that edge. So arranged, the three squares necessarily overlap, so the nut at each of the three points in the triangle must be counted twice, once for each square it participates in. Seen that way, the image is a visual representation of Pythagora's theorem for a triangle with sides 3, 4 and 5, with squares 9, 16 and 25, where 9 + 16 = 25 (so the bottom edge is the hypotenuse). On the other hand, the person who commented on the blog interpreted the number of nuts as ordinals, denoting the position of vertices in three lattices, represented by the "squares". The same person therefore interpreted e.g. the three nuts on the left edge of the triangle as standing for vertices indexed by ordinals 0, 1 and 2, and so representing a square lattice of side "2"; and so on for the other "squares". Seen that way, the image is a visual representation of a triangle with sides 2, 3 and 4, where the squares of the sides are 4, 9 and 16, where 4 + 9 ≠ 16 and so the triangle is obtuse rather than right (so the lower side is no longer the hypotenuse, since its square is no longer the sum of the squares of the other two sides) and the image is not a correct visual representation of Pythagoras' theorem. I want to say that even having written down the latter interpretation it still sounds deliberately obtuse to me, but the real lesson I think is that there are always multiple interpretations of the same statement, or formula, etc, and it's useful to be able to see as many of them as possible. At the same time, there is usually one intended interpretation and that's the one that should be preferred. All that is formalised in First Order Logic, in the concept of an interpretation, that is an assignment of truth values (true or false) to all the atoms of a predicate. A FOL interpretation is uniquely identified by the set of atoms to which it assigns the value true, therefore the number of possible interpretations is equal to the cardinality of the powerset of the set of atoms of a predicate. That's a lot of a interpretations! That's why you need an intended interpretation (also a concept in FOL). On the other hand, I might be wrong. Maybe it is deliberately obtuse to count the surface of a square by ordinals, rather than cardinals. Well I don't know. Bottom line is that in art, like in maths, one must always look for more than one way to see things and not assume that they know all the answers before they have asked all the questions.

CRConrad 2y ago

> I feel like this could all be helped if in casual counting we started at zero So when you've counted to one you've actually counted to two? I feel like some people have been damaged by C-like languages.

aaplok 3y ago

To count the length of the segment, we count the number of unit segments that compose it. Start with 1 (the first segment) and count up to 4 (the fourth segment), yielding a length of 4. We're consistent with the units: we are adding up lengths to get a length (metres, yards, or whatever). Counting (ie, adding up) points gives a number of points, which isn't a unit for lengths. Starting the enumeration from zero is a hack to recover the previous process of adding up unit lengths. This hack only works in this specific context of conversion. If you want to count points (say, the number of corners in the triangle) you'd need to start from 1.

sparky_z 3y ago

I'm sorry, but your suggestion doesn't make any sense to me. Saying "I have zero coconuts" would mean that you have a coconut? Would you have to say you had "negative 1" coconuts if you didn't have any? At a sandwich shop you would have to order a "no feet long" sub? And this is your plan to prevent confusion? All that would really do swap the meaning of a bunch of words around so that "zero" means "one", "one" means "two", etc. Then we'd have to call it a "Two three four" right triangle but remember to make it with "three" (four), "four" (five), and "five" (six) stones and we're right back where we started. The problem here is confusion about what quantity is actually being counted: the "fence posts" or the "fence lengths"? That's always going to depend on context - the speaker and listener have to be on the same page. There's no way to fix that by changing the number we count from.

baerrie 3y ago
Where do I say anything about zero becoming 1? What I am talking about is that counting arbitrary periods in a given quantity always starts with zero whether you count it or not. If you ground all of those coconuts up and decided to count how many cups of coconut you have in that mass, well you would be starting with an empty cup measure, which is zero. It’s not just fence posts
- lupire 3y ago
  Lok up "ordinal" and "offset". They are two different concepts that cannot be merged. Always using offsets will just cause confusing elsewhere.
  - YeGoblynQueenne 2y ago
    It's "ordinal" and "cardinal" really. If you wanna be all mathsy and stuff.
    - YeGoblynQueenne 2y ago
      Sorry about the above comment. I'm being way too snarky in this thread. I hope the following makes up for the snark in my other comments. What I meant above is that, in set theory, there are ordinal numbers and cardinal numbers, both of which are natural numbers. The easiest way to understand them is in terms of arrays: the number of elements in an array is a cardinal whereas an index into the array is an ordinal. Ordinals start at 0, because that's the first natural number, and that's prooobably? why array indexing traditionally begins at 0 (I'm totally guessing). Cardinals also start at 0. So, 0 is the "cardinality" of the empty set, {}, which has no elements and so no ordinal. In the set {+}, 0 is the ordinal that corresponds to the position of the single element of the set, in the set {+,+} 1 is the ordinal for the last element in the set etc. How this is relevant to the article above is that the artist very clearly intended the number of nuts in the "squares" at the sides of the triangle to be understood as cardinals: the squares represent sets and the cardinality of each set corresponds to the surface of that square. Sets are not ordered so the artist is free to place their elements in any arrangement, including a square, and placing a set in the form of a square on the side of an edge of the triangle clearly signals that its cardinality corresponds to the square of that edge. So arranged, the three squares necessarily overlap, so the nut at each of the three points in the triangle must be counted twice, once for each square it participates in. Seen that way, the image is a visual representation of Pythagora's theorem for a triangle with sides 3, 4 and 5, with squares 9, 16 and 25, where 9 + 16 = 25 (so the bottom edge is the hypotenuse). On the other hand, the person who commented on the blog interpreted the number of nuts as ordinals, denoting the position of vertices in three lattices, represented by the "squares". The same person therefore interpreted e.g. the three nuts on the left edge of the triangle as standing for vertices indexed by ordinals 0, 1 and 2, and so representing a square lattice of side "2"; and so on for the other "squares". Seen that way, the image is a visual representation of a triangle with sides 2, 3 and 4, where the squares of the sides are 4, 9 and 16, where 4 + 9 ≠ 16 and so the triangle is obtuse rather than right (so the lower side is no longer the hypotenuse, since its square is no longer the sum of the squares of the other two sides) and the image is not a correct visual representation of Pythagoras' theorem. I want to say that even having written down the latter interpretation it still sounds deliberately obtuse to me, but the real lesson I think is that there are always multiple interpretations of the same statement, or formula, etc, and it's useful to be able to see as many of them as possible. At the same time, there is usually one intended interpretation and that's the one that should be preferred. All that is formalised in First Order Logic, in the concept of an interpretation, that is an assignment of truth values (true or false) to all the atoms of a predicate. A FOL interpretation is uniquely identified by the set of atoms to which it assigns the value true, therefore the number of possible interpretations is equal to the cardinality of the powerset of the set of atoms of a predicate. That's a lot of a interpretations! That's why you need an intended interpretation (also a concept in FOL). On the other hand, I might be wrong. Maybe it is deliberately obtuse to count the surface of a square by ordinals, rather than cardinals. Well I don't know. Bottom line is that in art, like in maths, one must always look for more than one way to see things and not assume that they know all the answers before they have asked all the questions.